Kontsevich-Penner Model Research Articles

Random matrix, singularities and open/close intersection numbers

The s-point correlation function of a Gaussian Hermitian random matrix model, with an external source tuned to generate a multi-critical singularity, provides the intersection numbers of the moduli space for the p-th spin curves through a duality. For one marked point, the intersection numbers are expressed to all orders in the genus by Bessel functions. The matrix models for the Lie algebras of and provide the intersection numbers of non-orientable surfaces. The Kontsevich–Penner model, and the higher p-th Airy matrix model with a logarithmic potential, are investigated for the open intersection numbers, which describe the topological invariants of non-orientable surfaces with boundaries. String equations for the open/closed Riemann surface are derived from the structure of the s-point correlation functions. The Gromov–Witten invariants of the model are evaluated for one marked point as an application of the present method.

Open intersection numbers, Kontsevich-Penner model and cut-and-join operators

We continue our investigation of the Kontsevich--Penner model, which describes intersection theory on moduli spaces both for open and closed curves. In particular, we show how Buryak's residue formula, which connects two generating functions of intersection numbers, appears in the general context of matrix models and tau-functions. This allows us to prove that the Kontsevich--Penner matrix integral indeed describes open intersection numbers. For arbitrary $N$ we show that the string and dilaton equations completely specify the solution of the KP hierarchy. We derive a complete family of the Virasoro and W-constraints, and using these constraints, we construct the cut-and-join operators. The case $N=1$, corresponding to open intersection numbers, is particularly interesting: for this case we obtain two different families of the Virasoro constraints, so that the difference between them describes the dependence of the tau-function on even times.

On an Airy matrix model with a logarithmic potential

The Kontsevich–Penner model, an Airy matrix model with a logarithmic potential, may be derived from a simple Gaussian two-matrix model through a duality. In this dual version, the Fourier transforms of the n-point correlation functions can be computed in the closed form. Using Virasoro constraints, we find that in addition to the parameters tn, which appear in the Korteweg–de Vries hierarchies, one needs to introduce half-integer indices tn/2. The free energy as a function of those parameters may be obtained from these Virasoro constraints. The large N limit follows from the solution to an integral equation. This leads to explicit computations for a number of topological invariants.

Complex matrix model duality

The same complex matrix model calculates both tachyon scattering for the $c=1$ noncritical string at the self-dual radius and certain correlation functions of operators which preserve half the supersymmetry in $\mathcal{N}=4$ super-Yang-Mills theory. It is dual to another complex matrix model where the couplings of the first model are encoded in the Kontsevich-like variables of the second. The duality between the theories is mirrored by the duality of their Feynman diagrams. Analogously to the Hermitian Kontsevich-Penner model, the correlation functions of the second model can be written as sums over discrete points in subspaces of the moduli space of punctured Riemann surfaces.

BGWM as second constituent of complex matrix model

Earlier we explained that partition functions of various matrix models can be constructed from that of the cubic Kontsevich model, which, therefore, becomes a basic elementary building block in "M-theory" of matrix models. However, the less topical complex matrix model appeared to be an exception: its decomposition involved not only the Kontsevich tau-function but also another constituent, which we now identify as the Brezin-Gross-Witten (BGW) partition function. The BGW tau-function can be represented either as a generating function of all unitary-matrix integrals or as a Kontsevich-Penner model with potential 1/X (instead of X^3 in the cubic Kontsevich model).

Multicut solutions of the matrix Kontsevich-Penner model

Multicut solutions of the Hermitian one-matrix model parametrized by the recently introduced matrix model [1] with external field and Lagrangian having the form tr(ΛXΛX)-αN(log(1+X)−X) are considered. A brief review of the model, which describes the discretized moduli space of Riemann surfaces, is given. The general structure of multicut solutions is investigated, and it is shown that there arises an additional symmetry and thats parameters remain free for the (s+1)-cut solution. A detailed analysis of the one-cut solution is made. Among other results, all solutions of Kazakov type are reproduced. We also discuss the general form for the two-cut solution which arises as generalization of the string equation to the case of two cuts. The entire treatment is given in the approximation of planar diagrams.